The Riemann Problem And Interaction Of Waves For A Chromatography Model

This paper is concerned with the Riemann problem and interaction of waves for a chromatography model.Firstly,we analyze the solutions to the Riemann problem and elementary waves that possibly appear for the chromatography system.Especially,this chromatography model exhibiting a kind of nonclassical waves called singular shocks.Then,when the initial data are taken to be three piecewise constant states,we can construct the global solutions to the perturbed Riemann problem during the process of discussing the wave interaction problems in detail.In addition,it can be observed that the Riemann solutions are stable under small perturbations of the Riemann initial data.Finally,we study the generalized Riemann problem for this chromatography system.It can be proved that the structure of the solutions to the generalized Riemann problem is similar to the corresponding Riemann problem.And the solutions are locally unique in the class of piecewise C1 functions in a neighborhood of the origin.The paper is arranged as follows.In Chapter 1,we mainly introduce the background and some developments of the chromatography system.Then we give the derivation and simplification of the chromatography model under consideration.In Chapter 2,we present some basic concepts of the hyperbolic conservation laws and the solutions to the one-dimensional Riemann problem as the basic knowl-edge of our study,which will be helpful to the further discussion.In Chapter 3,we consider the Riemann problem for the chromatography equa-tions in a conservative form.We analyze the solutions and elementary waves of the Riemann problem for the chromatography system.Especially,this chromatography model exhibiting a kind of nonclassical waves called singular shocks,which do not satisfy the general Rankine-Hugoniot condition and entropy condition.The singular shocks is obtained through the method of Dafermos-DiPerna regularization to show the existence of a viscous solutionIn Chapter 4,we study the wave interaction problems.The global solution is obtained under the assumptions that the initial data are taken to be three piecewise constant states.The wave interaction problems are discussed in detail during the process of constructing global solutions to the perturbed Riemann problem.In addition,it can be observed that the Riemann solutions are stable under small perturbations of the Riemann initial dataIn Chapter 5,we study the generalized Riemann problem for this chromatog-raphy system.The unique local solution in the class of piecewise C1 functions in a neighborhood of the origin is obtained.This solution has a structure similar to the corresponding Riemann problem,i.e.,the Riemann solutions are stable.Especially,the singular shock is also stable.